Abstract
This investigation is concerned with the plane strain deformation of an infinite slab, containing a circular cavity, within the theory of finite elastostatics for a particular homogeneous isotropic compressible material, the so-called Blatz-Ko material. The body is subjected to uniform pressure, either internal or external. Exact closed-form solutions for the axisymmetric deformation and stress fields are obtained. In the case of internal pressure, it is found that the applied pressure may not exceed a certain maximum value p max. At a value of pressure p e (<p max), the governing equations lose ellipticity at the cavity wall. For greater values of pressure this solution remains smooth, though involving both elliptic and non-elliptic regions. Non-existence of axisymmetric solutions with discontinuous strain fields is established. The possibility of bifurication into a surface mode is considered and it is shown that this occurs at a value of pressure slightly smaller than p e. Such surface wrinkling leads to a periodic distribution of points of stress concentration, from which shear bands may initiate.
Similar content being viewed by others
References
A.E. Green and W. Zerna, Theoretical Elasticity, Oxford (1968).
P.J. Blatz and W.L. Ko, Application of finite elastic theory to the deformation of rubbery materials. Transactions of the Society of Rheology 6 (1962) 223–251.
J.K. Knowles and E. Sternberg, On the ellipticity of the equations of nonlinear elastostatics for a special material. Journal of Elasticity 5 (1975) 341–361.
R. Abeyaratne and C.O. Horgan, The pressurized hollow sphere problem in finite elastostatics for a class of compressible materials. International Journal of Solids and Structures 20 (1984) 715–723.
J.K. Knowles and E. Sternberg, On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics. Journal of Elasticity 8 (1978), pp. 329–379.
R. Abeyaratne, Discontinuous deformation gradients in plane finite elastostatics of incompressible materials. Journal of Elasticity 10 (1980) 255–293.
J.R. Rice, The localization of plastic deformation. Proceedings of the 14th IUTAM Congress, W.T. Koiter (ed), Delft, the Netherlands, August 1976 207–220. North-Holland, Amsterdam, (1977).
S.P. Timoshenko and J.N. Goodier, Theory of Elasticity, McGraw-Hill (1970).
J.K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain. Archive for Rational mechanics and Analysis 63 (1977) 321–336.
R. Abeyaratne, Discontinuous deformation gradients in the finite twisting of an incompressible elastic tube. Journal of Elasticity 11 (1981) 43–80.
J.K. Knowles and E. Sternberg, Discontinuous deformation gradients near the tip of a crack in finite anti-plane shear: An example. Journal of Elasticity 10 (1980) 81–110.
R. Abeyaratne, Discontinuous deformation gradients away from the tip of a crack in anti-plane shear. Journal of Elasticity 11 (1981) 373–393.
G.F. Fowler, Finite plane and anti-plane elastostatic fields with discontinuous deformation gradients near the tip of a crack. Journal of Elasticity 14 (1984) 287–328.
M.A. Biot, Mechanics of Incremental Deformation, Wiley, New York (1965).
J.L. Bassani, D. Durban and J.W. Hutchinson, Bifurcation at a spherical hole in an infinite elastoplastic medium. Mathematical Proceedings of the Cambridge Philosophical society 87 (1980) 339–356.
B.D. Reddy, The occurrence of surface instabilities and shear bands in plane strain deformation of an elastic half-space. Quarterly Journal of Mechanics and Applied Mathematics 36 (1983) 337–350.
V. Tvergaard, A. Needleman and K.K. Lo, Flow localization in the plane strain tensile test. Journal of the Mechanics and Physics of Solids 29 (1981) 115–142.
M. Larsson, A. Needleman, V. Tvergaard and B. Storåkers, Instability and failure of internally pressurized ductile metal cylinders. Journal of the Mechanics and Physics of Solids 30 (1982) 121–154.
N. Triantafyllidis, A. Needleman and V. Tvergaard, On the development of shear bands in pure bending. International Journal of Solids and Structures 18 (1982) 121–138.
Author information
Authors and Affiliations
Additional information
This work was supported by the U.S. Army Research Office under Grant DAAG29-83-K-0145 (R.A. & C.O.H.) and by the U.S. National Science Foundation under Grant MEA 78-26071 (C.O.H.).
Rights and permissions
About this article
Cite this article
Abeyaratne, R., Horgan, C.O. Initiation of localized plane deformations at a circular cavity in an infinite compressible nonlinearly elastic medium. J Elasticity 15, 243–256 (1985). https://doi.org/10.1007/BF00041423
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00041423